ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexv Unicode version

Theorem rexv 2748
Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
rexv  |-  ( E. x  e.  _V  ph  <->  E. x ph )

Proof of Theorem rexv
StepHypRef Expression
1 df-rex 2454 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ( x  e. 
_V  /\  ph ) )
2 vex 2733 . . . 4  |-  x  e. 
_V
32biantrur 301 . . 3  |-  ( ph  <->  ( x  e.  _V  /\  ph ) )
43exbii 1598 . 2  |-  ( E. x ph  <->  E. x
( x  e.  _V  /\ 
ph ) )
51, 4bitr4i 186 1  |-  ( E. x  e.  _V  ph  <->  E. x ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1485    e. wcel 2141   E.wrex 2449   _Vcvv 2730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-rex 2454  df-v 2732
This theorem is referenced by:  rexcom4  2753  spesbc  3040  abnex  4432  dfco2  5110  dfco2a  5111
  Copyright terms: Public domain W3C validator